Algorithms for the line-constrained disk coverage and related problems

Given a set P of n points and a set S of m disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of P. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of P by a line ℓ. We present an O((n+m)log⁡(n+m)) time algorithm for the problem. This improves the previously best result of O(nm+nlog⁡n) time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of S are located on a line ℓ while points of P can be anywhere in the plane. Our algorithm runs in O((n+m)log⁡(m+n)+mlog⁡mlog⁡n) time, which improves the previously best result of O(nmlog⁡(m+n)) time. In addition, our results lead to an algorithm of O(n3log⁡n) time for a half-plane coverage problem (given n half-planes and n points, find a smallest subset of half-planes covering all points); this improves the previously best algorithm of O(n4log⁡n) time. Further, if all half-planes are lower ones, our algorithm runs in O(nlog⁡n) time while the previously best algorithm takes O(n2log⁡n) time.

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.

Geometric Covering and Packing problems have been extensively studied in the last few decades and have applications in diverse areas. Several variants and generalizations of these problems have been studied recently. In this paper, we look at the following covering variants where we require that each point is “uniquely” covered, i.e., it is covered by exactly one object: Unique Coverage problem, where we want to maximize the number of uniquely covered points and Exact Cover problem, where we want to uniquely cover every point and minimize the number of objects used for covering. We also look at the following generalizations: Multi Cover problem, a generalization of Set Cover, the objective is to select the minimum subset of objects with the constraint that each input point p is covered by at least $$d_p$$ objects in the solution, where $$d_p$$ is the demand of point p. And Shallow Packing problem, a generalization of Packing problem, where we want to select the maximum subset of objects with the constraint that any point in the plane is contained in at most k objects in the solution. The above problems are NP-hard even for unit squares in the plane. Thus, the focus has been on obtaining good approximation algorithms. Local search has been quite successful in the recent past in obtaining good approximation algorithms for a wide variety of problems. We consider the Unique Coverage and Multi Cover problems on non-piercing objects, which is a broad class that includes squares, disks, pseudo-disks, etc. and show that the local search algorithm yields a PTAS approximation under the assumption that the depth of every input point is at most some fixed constant. For Unique Coverage we further assume that every object has at most a constant degree. For the Shallow Packing problem, we show that the local search algorithm yields a PTAS approximation for objects with sub-quadratic union complexity, which is a very broad class of objects that even includes non-piercing objects. For the Exact Cover problem, we show that finding a feasible solution is NP-hard even for unit squares in the plane, thus negating the existence of polynomial time approximation algorithms.

A sequence of exact algorithms to solve the Vertex Cover and Maximum Independent Set problems have been proposed in the literature. All these algorithms appeal to a very conservative analysis that considers the size of the search tree, under a worst-case scenario, to derive an upper bound on the running time of the algorithm. In this paper we propose a different approach to analyze the size of the search tree. We use amortized analysis to show how simple algorithms, if analyzed properly, may perform much better than the upper bounds on their running time derived by considering only a worst-case scenario. This approach allows us to present a simple algorithm of running time O(1.194kk2 + n) for the parameterized Vertex Cover problem on degree-3 graphs, and a simple algorithm of running time O(1.1255n) for the Maximum Independent Set problem on degree-3 graphs. Both algorithms improve the previous best algorithms for the problems.

A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares

A 4.31-approximation for the geometric unique coverage problem on unit disks

Under the word RAM model, we design three data structures that can be constructed in O(n √{lg n}) time over n points in an n × n grid. The first data structure is an O(n lg^ε n)-word structure supporting orthogonal range reporting in O(lg lg n+k) time, where k denotes output size and ε is an arbitrarily small constant. The second is an O(n lg lg n)-word structure supporting orthogonal range successor in O(lg lg n) time, while the third is an O(n lg^ε n)-word structure supporting sorted range reporting in O(lg lg n+k) time. The query times of these data structures are optimal when the space costs must be within O(n polylog n) words. Their exact space bounds match those of the best known results achieving the same query times, and the O(n √{lg n}) construction time beats the previous bounds on preprocessing. Previously, among 2d range search structures, only the orthogonal range counting structure of Chan and Pǎtraşcu (SODA 2010) and the linear space, O(lg^ε n) query time structure for orthogonal range successor by Belazzougui and Puglisi (SODA 2016) can be built in the same O(n √{lg n}) time. Hence our work is the first that achieve the same preprocessing time for optimal orthogonal range reporting and range successor. We also apply our results to improve the construction time of text indexes.

More on change-making and related problems

We consider the following natural "above guarantee" parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study the related problem Exact Detour that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k, and a deterministic algorithm with running time about 6.745^k, showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs.

We consider the following general scheduling problem studied recently by Moseley. There are $n$ jobs, all released at time $0$, where job $j$ has size $p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its completion time. The goal is to find a schedule on $m$ machines with minimum total cost. We give an $O(1)$ approximation for the problem, improving upon the previous $O(\log \log nP)$ bound ($P$ is the maximum to minimum size ratio), and resolving the open question of Moseley. We first note that the scheduling problem can be reduced to a clean geometric set cover problem where points on a line with arbitrary demands, must be covered by a minimum cost collection of given intervals with non-uniform capacity profiles. Unfortunately, current techniques for such problems based on knapsack cover inequalities and low union complexity, completely lose the geometric structure in the non-uniform capacity profiles and incur at least an $\Omega(\log\log P)$ loss. To this end, we consider general covering problems with non-uniform capacities, and give a new method to handle capacities in a way that completely preserves their geometric structure. This allows us to use sophisticated geometric ideas in a black-box way to avoid the $\Omega(\log \log P)$ loss in previous approaches. In addition to the scheduling problem above, we use this approach to obtain $O(1)$ or inverse Ackermann type bounds for several basic capacitated covering problems.

Covering problems for partial words and for indeterminate strings

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a global simple multiobjective evolutionary algorithm (GSEMO) and apply it on a minimum weight general cover problem, which is general enough to subsume many important problems including the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is nonsubmodular. We show that GSEMO yields theoretically guaranteed approximation ratios matching those achievable by a greedy algorithm in expected polynomial time when the potential function g is polynomial in the input size and the minimum gap between different g-values is a constant. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by National Natural Science Foundation of China [11771013, U20A2068]; Zhejiang Provincial Natural Science Foundation of China [LD19A010001].

In this paper, we study the target coverage problem of directed sensor networks with rotatable angles. The main purpose of this problem is to maximize number of targets to be covered while minimize number of activated directional sensors. Coverage problem is well-studied for omni-directional sensors, however, it is still a challenge task for directional sensor. The directed sensor nodes used in the previous works are composed of fixed number of sensors, and the orientations of sensors are fixed. In this study, we consider the directed sensors with arbitrary sensing orientations. Our algorithm constructs sectors of sensors based on targets within the sensing distance of sensors to gain higher coverage rate and lower ratio of active sensors. To evaluate the performance of proposed algorithm, several simulations are conducted. Experimental results show that our algorithm can contribute to increase the coverage rate of targets and decrease the ratio of activated directional sensors obviously than previous algorithms, thus gain better coverage performance and longer lifetime for directed sensor networks.

Static area coverage of the heterogeneous autonomous underwater vehicle (AUV) group is widely used in many fields. With the use of the centroidal Voronoi tessellation (CVT) algorithm, the coverage problem can be resolved. However, the CVT method, which is optimized with the location evaluation function, cannot consider the heterogeneity of AUVs when the group performs the static area coverage task and will cause a waste of resources. In this paper, considering different AUVs’ task requirements and detection capabilities comprehensively, we propose a coverage control optimization algorithm based on a biological competition mechanism (BCM). By using BCM, the task load of each AUV can be distributed consistently. In addition, we provide strict proof of the consistency of the algorithm based on the Lyapunov method. Simulation results demonstrate that with the proposed algorithm, the location distribution of the heterogeneous AUV group for area coverage is close to the balanced value, and the performance is better than the CVT algorithm for static area coverage.

We propose approximation algorithms under game-theoretic considerations. We indroduce and study the general covering problem which is a natural generalization of the well-studied max-n -cover problem. In the general covering problem, we are given a universal set of weighted elements E and n collections of subsets of the elements. The task is to choose one subset from each collection such that the total weight of their union is as large as possible. In our game-theoretic setting, the choice in each collection is made by an independent player. For covering an element, the players receive a payoff defined by a non-increasing utility sharing function. This function defines the fraction that each covering player receives from the weight of the elements. We show how to construct a utility sharing function such that every Nash Equilibrium approximates the optimal solution by a factor of $1-{{1} \over {e}}$. We also prove that any sequence of unilateral improving steps is polynomially bounded. This gives rise to a polynomial-time local search approximation algorithm whose approximation ratio is best possible.